Power statistics-equivalent finite groups

This article defines an equivalence relation over the collection of groups. View a complete list of equivalence relations on groups.

Definition

Two finite groups are termed power statistics-equivalent finite groups if their power statistics functions are identical.

Facts

Relation with group properties

Note that the third column (first question column) is the conjunction of the fourth and fifth columns.

Group property Finite version Is any group power statistics-equivalent to a group with this property isomorphic to it? Does any group power statistics-equivalent to a group with this property also have this property? Are any two groups with this property that are power statistics-equivalent also isomorphic?
Cyclic group Finite cyclic group Yes Yes Yes
Abelian group Finite abelian group No No Yes: Finite abelian groups with the same power statistics are isomorphic
Nilpotent group Finite nilpotent group No  ? No

=Relation with other relations

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